Analytic potential expression of a three-plane parallel-electrode system with special apertures

Abstract

Potential distributions in all the space of a two-dimensional lens consisting of three parallel plates of symmetrical construction and specific apertures are obtained analytically by finding special transformation functions between the real lens space and the corresponding imaginary one. The functions contain two arbitrary constantsmandnconnecting the inner aperture 2Yi, the outer one 2Yo, and the distancedbetween the electrodes. The distancedis given byn\pi/2. Y_{in}=Y_{i}/dandY_{on} = Y_{o}/dare functions of the ratiom/nonly. Therefore, the transformation functions can be applied to specific combinations of Yin, Yon, andd. Yin= Yon= 0.6789884 is obtained form/n= 0.1851789. As an example, a full mapping of the lines of potential and electric field form = 1/4, n = 2, andd = \piis shown. In this case one obtains Yin= 0.6406773 and Yon= 0.5507097.